Midterm Test

			Marking Scheme and TA Comments


Question 1
==========

   Most of the students did well on Question 1. 

   In addition to incorrect calculations, common errors included: 
      - not rounding-to-nearest correctly (deducted 1 mark), and 
      - not computing gradual underflow correctly (deducted 1 mark).


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Question 2
==========

Part (a): 

Marking scheme
   2 pts for identifying r1 as the inaccurate root, 
   2 points for identifying that the cause of the inaccuracy is the
     catastrophic cancellation that occurs when computing 
     - b + sqrt(b^2-4ac) 
   1 point for a reasonable analysis of the size of the error

Comments
   Students lost points because they identified catastrophic cancellation 
   as -b-sqrt(b^2-4ac), which they claimed is close to -2b when b is large 
   and the 4ac part is essentially lost. This is all true but it's not the 
   cause of the problem. The root r2 is calculated this way but it is 
   numerically accurate. 

   Another incorrect identification of catastrophic cancellation is the 
   calculation of b^2-4ac.  While this could occur for other values of a, 
   b, and c (specifically when b^2 ~= 4ac), this doesn't happen here,
   because b^2 is very large and 4ac is small in comparison. Some
   students wrote that this causes the 4ac to be "lost" in computation as 
   if it were not there. This is all true, but doesn't affect r2, the accurate 
   root, so it is not the cause of the inaccuracies.

   If the students identified incorrect sources of error and also the 
   correct one, then a point was taken off.

   Some students also suggested that the sqrt() function is ill-conditioned. 
   Calculating the condition number (xf'(x)/f(x)) would show that it is 1/2, 
   and hence well-conditioned.


Part (b)

Marking scheme
  3 pts for using either the formula a * r1 * r2 = c or the similar one 
    in Heath's textbook correctly 
  2 points for reasoning/explanation

Comments
   A very common incorrect use of the formula is for both roots. Since this 
   is no improvement over the formula given on the question (you still have 
   one inaccurate root and one accurate root), no points were awarded for this
   answer.

   Additionally, some students, who have identified the catastrophic 
   cancellation as -b-sqrt(b^2-4ac), leading them to identify
   r2 as the inaccurate root, tried to "fix" the computation for r2. 
   If they used Heath's formula, no points were awarded,
   because they introduced the cancellation error for r2 when there was none
   to begin with. However, if they used the relation a*r1*r2 = c, then I
   awarded points (as we discussed) because it doesn't make sense to penalize
   them again for their mistake in part (a).


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Question 3
==========

   Most students did quite well on this question.

   A few students forgot to take absolute values of the components when 
   calculating norms.

   A few students made careless computation mistakes.

   For each incorrect answer, I took off 1 point.


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Question 4
==========

Most students did well on this question. Some students, though,
solved Ux = b instead of Ux = \hat(b).